Energy in a spring-mass system

In summary, the question asks for the speed of a mass attached to a horizontal spring with stiffness 0.6 N/m when the spring returns to its relaxed length of 14 cm (0.14 m). Using the energy principle, the equation (1/2) * ks * sf^2 = [ (1/2) * m * vi^2 ] + [ (1/2) * ks * si^2 ] can be used to solve for the initial velocity (Vi). However, this equation may cause confusion as the work done by non conservative forces (W) is not taken into account. To solve for the speed, the equation sqrt[ { (1/2) * 0.6 * (0
  • #1
ryoonc
5
0

Homework Statement


A horizontal spring with stiffness 0.6 N/m has a relaxed length of 14 cm (0.14 m). A mass of 24 grams (0.024 kg) is attached and you stretch the spring to a total length of 23 cm (0.23 m). The mass is then released from rest. What is the speed of the mass at the moment when the spring returns to its relaxed length of 14 cm (0.14 m)?

Homework Equations


Using the Energy Principle:
Kf + Uf = Ki + Ui + W (Kf and W cancel out in this equation), therefore:
(1/2) * ks * sf^2 = [ (1/2) * m * vi^2 ] + [ (1/2) * ks * si^2 ]

The Attempt at a Solution


I've listed all the variables given:
ks = 0.6 N/m
Lo = 0.14 m
m = 0.024 kg
si = 0.09 m
sf = 0 m

Assuming that I'm looking for the speed of the mass when the spring is at its relaxed length (sf or Lo), I tried plugging in the variables to the energy principle, and got:

0 = [ (1/2) * 0.024kg * Vi^2 ] + [ (1/2) * 0.6 N/m * (0.09 m)^2 ]

However, this confuses me as I thought the question is asking for the final velocity, when the spring is at its relaxed position, and when I try solving this equation for Vi I get a negative number..

Ok so a friend just came by and solved it in 5 minutes, using this equation:
deltaU -> delta K
sqrt[ { (1/2) * 0.6 * (0.09^2) } / m ] = v

Could someone kindly explain the concept to me?

Thanks!
 
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  • #2
You made a few errors. When applying your conservation of energy equations, why did you say W cancels Kf? And why did you use a non zero value for Ki when it starts from rest? Note that "W" in your equation refers to work done by non conservative forces only. Are there any such forces actng?
 
  • #3
Wow you're fast, thanks for your reply Jay.

I said W cancels Kf because my book canceled them out, for whichever reason I don't know why. I used a non-zero value for Ki mainly because I didn't really know what I was doing, but I realize now that initial kinetic energy in this case is zero (maybe in other cases too I'm not sure), and potential energy is whatever number in joules, or in other words:

K + U = constant

So when kinetic energy is increasing, potential energy is decreasing.

I'm still working on the rest, but I feel pretty good about finally getting at least a vague idea on the concept of K and U.
 
  • #4
OK, but note that K+U remains constant only when non conservative forces such as weight and spring forces are acting. When other forces act, like friction, applied forces, etc., then K + U is not constant , and you mist use the general form of the energy equation which includes the work done by those non conservative forces.
 

1. What is a spring-mass system?

A spring-mass system is a physical system that consists of a mass attached to a spring. The mass can be moved up and down by compressing or stretching the spring, causing potential and kinetic energy to be stored and released.

2. What is the relationship between the spring constant and the frequency of oscillation?

The spring constant, also known as the force constant, is directly proportional to the frequency of oscillation in a spring-mass system. This means that as the spring constant increases, the frequency of oscillation also increases. Similarly, as the spring constant decreases, the frequency of oscillation decreases.

3. How does the amplitude affect the energy in a spring-mass system?

The amplitude, or maximum displacement, of the mass in a spring-mass system does not affect the total energy of the system. However, it does affect the potential and kinetic energy at any given point in time. As the amplitude decreases, the potential energy increases and the kinetic energy decreases, and vice versa.

4. What happens to the energy in a spring-mass system when damping is introduced?

When damping is introduced in a spring-mass system, the total energy of the system decreases over time due to the dissipation of energy through friction or other resistance. This causes the amplitude of the oscillations to decrease, eventually reaching equilibrium where there is no more energy left in the system.

5. How does the mass of the object affect the energy in a spring-mass system?

The mass of the object attached to the spring affects the total energy of the system. As the mass increases, the total energy increases as well. This is because a heavier mass requires more energy to be moved and oscillate compared to a lighter mass.

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